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In this paper, we prove that there exists a universal constant $C$, dependingonly on positive integers $n\geq 3$ and $p\leq n-1$, such that if $M^n$ is acompact free boundary submanifold of dimension $n$ immersed in the Euclideanunit ball $\mathbb{B}^{n+k}$ whose size of the traceless second fundamentalform is less than $C$, then the $p$th cohomology group of $M^n$ vanishes. Also,employing a different technique, we obtain a rigidity result for compact freeboundary surfaces minimally immersed in the unit ball $\mathbb{B}^{2+k}$.

Vanishing theorems for the cohomology groups of free boundary hypersurfaces